In the survey paper Poset Topology: Tools and Applications by Michelle Wachs, there is the following theorem on p46:
Theorem 3.2.4 (Björner and Wachs [40]). Suppose $P$ is a poset for which $\hat{P}$ admits an EL-labeling. Then $P$ has the homotopy type of a wedge of spheres, where the number of $i$-spheres is the number of decreasing maximal $(i+2)$-chains of $\hat{P}$.
The notation $\hat{P}$ means $ P \cup \{\hat{0}, \hat{1}\}$. The homotopy type of a poset $P$ is the homotopy type of the geometric realization of its order complex $\Delta(P)$.
Now let $P$ be a poset of order $n+1$ and length $0$ (i.e. just isolated points), then $\hat{P}$ admits an obvious EL-labeling with $n$ decreasing maximal $2$-chains. So by Theorem 3.2.4, $P$ should have the homotopy type of a wedge of $n$ number of $0$-sphere, i.e just one point, but it is not correct because it is a set of $n+1$ points.
I would need some clarifications about this inconsistency, in particular:.
Question: Is the statement of Theorem 3.2.4 correct if we assume that there is no isolated point?
Remark: If a poset admits an EL-labeling then it is Cohen-Macaulay (see here), and a poset is Cohen-Macaulay iff its Stanley–Reisner ring is a Cohen-Macaulay ring.